Question

Consider the linear function that is represented by the equation y = 4 x + 5 and the linear function that is represented by the table below.

x
y
2
16
4
26
6
36
8
46

Which statement is correct regarding their slopes and y-intercepts?
The function that is represented by the equation has a steeper slope and a greater y-intercept.
The function that is represented by the equation has a steeper slope, and the function that is represented by the table has a greater y-intercept.
The function that is represented by the table has a steeper slope, and the function that is represented by the equation has a greater y-intercept.
The function that is represented by the table has a steeper slope and a greater y-intercept.

Answer 1

To compare the slopes and y-intercepts of the two linear functions, we start by analyzing the given equation y = 4x + 5:

1. **Slope and Y-Intercept of the Equation:**
- The slope (m) of the equation y = 4x + 5 is 4.
- The y-intercept (b) is 5.

Now, we'll determine the slope and y-intercept of the function represented by the table:

### Data from the table:
- When x = 2, y = 16
- When x = 4, y = 26
- When x = 6, y = 36
- When x = 8, y = 46

### Calculating the slope of the table:
Using two points from the table, we can calculate the slope. Let's take points (2, 16) and (4, 26):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{26 - 16}{4 - 2} = \frac{10}{2} = 5
\]

### Y-Intercept of the Table Function:
To find the y-intercept, we can use the slope we calculated (5) and one of the points. We'll use the point (2, 16):

We know that:
\[
y = mx + b \implies 16 = 5(2) + b
\]
\[
16 = 10 + b \implies b = 6
\]

### Summary of both functions:
1. For the function represented by the equation y = 4x + 5:
- Slope: 4
- Y-Intercept: 5

2. For the function represented by the table:
- Slope: 5
- Y-Intercept: 6

### Comparison:
- The slope of the function from the table (5) is steeper than the slope of the equation (4).
- The y-intercept of the function from the table (6) is greater than the y-intercept of the equation (5).

### Conclusion:
The correct statement is:
**The function that is represented by the table has a steeper slope and a greater y-intercept.**